Systems and methods for obtaining information on a key in BB84 protocol of quantum key distribution

ABSTRACT

Systems and methods for obtaining information on a key in the BB84 (Bennett-Brassard 1984) protocol of quantum key distribution are provided. A representative system comprises a quantum cryptographic entangling probe, comprising a single-photon source configured to produce a probe photon, a polarization filter configured to determine an initial probe photon polarization state for a set error rate induced by the quantum cryptographic entangling probe, a quantum controlled-NOT (CNOT) gate configured to provide entanglement of a signal with the probe photon polarization state and produce a gated probe photon so as to obtain information on a key, a Wollaston prism configured to separate the gated probe photon with polarization correlated to a signal measured by a receiver, and two single-photon photodetectors configured to measure the polarization state of the gated probe photon.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority to co-pending U.S. provisionalapplication entitled, “Quantum Cryptographic Entangling Probe,” havingSer. No. 60/617,796, filed Oct. 9, 2004, which is incorporated herein byreference.

GOVERNMENT INTEREST

The invention described herein may be manufactured, used, and licensedby or for the United States Government.

BACKGROUND

1. Technical Field

The present disclosure is generally related to the secure communicationof encrypted data using quantum cryptography.

2. Description of the Related Art

Research efforts by many investigators have significantly advanced thefield of quantum cryptography since the pioneering discoveries ofWiesner, Bennett and Brassard, as shown in the following references: N.Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,”Rev. Mod. Phys. Vol. 74, pp. 145-195 (2002); S. Wiesner, “Conjugatecoding,” SIGACT News Vol. 15, No. 1, pp. 78-88 (1983); C. H. Bennett andG. Brassard, “Quantum cryptography, public key distribution and cointossing,” Proceedings of the IEEE International Conference on Computers,Systems, and Signal Processing, Bangalore, India, pp. 175-179, (IEEE1984); C. H. Bennett and G. Brassard, “Quantum public key distributionsystem,” IBM Tech. Discl. Bull. Vol. 28, No. 7, pp. 3153-3163, (1985),all of which are incorporated herein by reference in their entireties.Emphasis has been placed on quantum key distribution, the generation bymeans of quantum mechanics of a secure random binary sequence which canbe used together with the Vernam cipher (one-time pad) as discussed inG. Vernam, “Cipher printing telegraph systems for secret wire and radiotelegraph communications,” J. Am. Inst. Electr. Eng. Vol. 45, pp.295-301 (1926), which is incorporated herein by reference in itsentirety, for secure encryption and decryption. Various protocols havebeen devised for quantum key distribution, including the single-particlefour-state Bennett-Brassard protocol (BB84), Bennett (1984), thesingle-particle two-state Bennett protocol (B92) as in C. H. Bennett,“Quantum cryptography using any two nonorthogonal states,” Phys. Rev.Lett. Vol. 68, pp. 3121-3124 (1992), which is incorporated herein byreference in its entirety, and the two-particle entangled-stateEinstein-Podolsky-Rosen (EPR) protocol as in A. K. Ekert, “Quantumcryptography based on Bell's theorem,” Phys. Rev. Lett. Vol. 57, pp.661-663 (1991), which is incorporated herein by reference in itsentirety. However the original BB84 protocol is presently perceived asthe most practical and robust protocol.

One effective implementation of the BB84 protocol uses single photonslinearly polarized along one of the four basis vectors of two sets ofcoplanar orthogonal bases oriented at an angle of 45 degrees(equivalently, π/4) relative to each other. The polarization measurementoperators in one basis do not commute with those in the other, sincethey correspond to nonorthogonal polarization states. At a fundamentallevel, the potential security of the key rests on the fact thatnonorthogonal photon polarization measurement operators do not commute,and this results in quantum uncertainty in the measurement of thosestates by an eavesdropping probe, as in H. E. Brandt, “Positive operatorvalued measure in quantum information processing,” Am. J. Phys. Vol. 67,pp. 434-439 (1999), which is incorporated herein by reference in itsentirety. Before transmission of each photon, the transmitter andreceiver each independently and randomly select one of the two bases.The transmitter sends a single photon with polarization chosen at randomalong one of the orthogonal basis vectors in the chosen basis. Thereceiver makes a polarization measurement in its chosen basis. Next, thetransmitter and the receiver, using a public communication channel,openly compare their choices of basis, without disclosing thepolarization states transmitted or received. Events in which thetransmitter and the receiver choose different bases are ignored, whilethe remaining events ideally have completely correlated polarizationstates. The two orthogonal states in each of the bases encode binarynumbers 0 and 1, and thus a sequence of photons transmitted in thismanner can establish a random binary sequence shared by both thetransmitter and the receiver and can then serve as the secret key,following error correction and privacy amplification, as in C. H.Bennett, G. Brassard, C. Crepeau, and V. M. Maurer, “Generalized privacyamplification,” IEEE Trans. Inf. Theor, Vol. 41, pp. 1915-1923 (1995),and C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin,“Experimental quantum cryptography,” J. Cryptology, Vol. 5, pp. 3-28(1992), both of which are incorporated herein by reference in theirentireties. Privacy amplification is of course necessary because of thepossibility of an eavesdropping attack, as in Gisin (2002), Bennett(1984), and Bennett (1985). Using the Vernam cipher, the key can then beused to encode a message which can be securely transmitted over an opencommunication line and then decoded, using the shared secret key at thereceiver. (The encrypted message can be created at the transmitter byadding the key to the message and can be decrypted at the receiver bysubtracting the shared secret key.)

Numerous analyses of various eavesdropping strategies have appeared inthe literature, see e.g., Gisin (2002). Attack approaches includecoherent collective attacks in which the eavesdropper entangles aseparate probe with each transmitted photon and measures all probestogether as one system, and also coherent joint attacks in which asingle probe is entangled with the entire set of carrier photons.However, these approaches require maintenance of coherent superpositionsof large numbers of states.

SUMMARY

Systems and methods for obtaining information on the BB84 protocol ofquantum key distribution are provided. In this regard, an embodiment ofa system can be implemented as follows. A single photon source isconfigured to produce a probe photon. A polarization filter isconfigured to determine an optimum initial probe photon polarizationstate for a set error rate induced by the device. A quantumcontrolled-NOT (CNOT) gate is configured to provide entanglement of asignal photon polarization state with the probe photon polarizationstate and produce a gated probe photon correlated with the signal photonso as to obtain information on a key. A Wollaston prism is configured toseparate the gated probe photon with polarization appropriatelycorrelated with a signal photon as measured by a receiver. Twosingle-photon photodetectors are configured to measure the polarizationstate of the gated probe photon. The CNOT gate may be further configuredto provide optimum entanglement of the signal with the probe photonpolarization state so as to obtain maximum Rényi information from thesignal.

An embodiment of a method for obtaining information on a key in the BB84protocol of quantum key distribution comprises the steps of: configuringa single photon source for producing a probe photon; determining aninitial probe photon polarization state corresponding to a set errorrate induced by a probe; entangling a signal with a probe photonpolarization state and producing a gated probe photon; separating thegated probe photon with polarization correlated with a signal measuredby a receiver; measuring the polarization state of the gated probephoton; accessing information on polarization-basis selection availableon a public classical communication channel between the transmitter andthe receiver; and determining the polarization state measured by thereceiver. A quantum CNOT gate may be further configured to provideoptimum entanglement of the signal with the probe photon polarizationstate so as to obtain maximum Rényi information from the signal.

Other systems, methods, features, and advantages of the presentdisclosure will be or become apparent to one with skill in the art uponexamination of the following drawings and detailed description. It isintended that all such additional systems, methods, features, andadvantages be included within this description, be within the scope ofthe present disclosure, and be protected by the accompanying claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the disclosure can be better understood with referenceto the following drawings. The components in the drawings are notnecessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the present disclosure. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1 is a schematic diagram of an embodiment of a system utilizing aquantum cryptographic entangling probe.

FIG. 2 is flowchart of an embodiment of a method for obtaininginformation on a key in BB84 protocol of quantum key distributionincorporating an embodiment of a quantum cryptographic entangling probe.

FIG. 3 is a schematic diagram of another embodiment of a system forobtaining information on a key in BB84 protocol of quantum keydistribution incorporating an embodiment of a quantum cryptographicentangling probe.

DETAILED DESCRIPTION

Reference is now made in detail to the description of several exemplaryembodiments as illustrated in the drawings. The disclosure may, however,be embodied in many different forms and should not be construed aslimited to the embodiments set forth herein; rather, these embodimentsare intended to convey the scope of the disclosure to those skilled inthe art. Furthermore, all “examples” given herein are intended to benon-limiting.

The present disclosure provides systems and methods for obtaininginformation on a key in the BB84 protocol of quantum key distribution.Some embodiments can be implemented as quantum cryptographic entanglingprobes for eavesdropping on the BB84 protocol.

FIG. 1 shows a system 100 utilizing a quantum cryptographic entanglingprobe 300 to obtain information from a transmitted signal. The quantumcryptographic entangling probe 300 is presented in greater detail in thediscussion regarding FIG. 3 below. The transmitter 110 sends a signalphoton 140 (an incident photon) through a quantum channel 130. Thequantum channel 130 is an optical pathway and may be optical fiber orairspace, as non-limiting examples. The quantum cryptographic entanglingprobe 300 entangles a probe photon (not shown) with the transmittedsignal photon 140 to produce a gated probe photon 150 and a gated signalphoton 160. The gated signal photon 160 is relayed to the receiver 120,while the gated probe photon 150 is utilized by the quantumcryptographic entangling probe 300 to determine the state that will mostlikely be measured by the receiver 120 in response to receiving thegated signal photon 160.

FIG. 2 is a flowchart depicting an embodiment of a method for obtaininginformation on a key in the BB84 protocol of quantum key distribution,and thus for determining the state most likely measured by a receiver. Asignal photon 140, is received from a transmitter 110 in step 210. Asingle-photon source 320 (see FIG. 3) is configured for producing aprobe photon in step 220. The optimum probe photon polarization state,corresponding to a set error rate induced by the quantum cryptographicentangling probe 300, is determined in step 230. In step 240, the signalphoton 140 received in step 210 is entangled with the probe photon fromstep 230 to produce a gated probe photon 150. The gated probe photon 150is separated with polarization correlated to a signal measured by areceiver in step 250. Step 260 shows that the polarization state of thegated probe photon 150 is measured. Finally, the information onpolarization-bases selection utilized by the transmitter and receiver isaccessed on a public classical communication channel in step 270 so thatthe polarization state measured by the receiver 120 may be determined instep 280.

For the standard four-state (BB84) protocol, Bennett (1984), of quantumkey distribution in quantum cryptography, an eavesdropping probeoptimization was performed, by B. A. Slutsky, R. Rao, P. C. Sun, and Y.Fainman, “Security of quantum cryptography against individual attacks,”Phys. Rev. A Vol. 57, pp. 2383-2398 (1998), which is incorporated hereinby reference in its entirety, which on average yields the mostinformation to the eavesdropper for a given error rate caused by theprobe. The most general possible probe consistent with unitarity wasconsidered in which each individual transmitted bit is made to interactwith the probe so that the carrier and the probe are left in anentangled state, and measurement by the probe, made subsequent tomeasurement by the legitimate receiver, yields information about thecarrier state. The probe can be used in an individual attack in whicheach transmitted photon is measured independently, given that thepolarization basis is revealed on the public communication channel. Seethe following references: Slutsky (1998), as above; C. A. Fuchs and A.Peres, “Quantum-state disturbance versus information gain: uncertaintyrelations for quantum information,” Phys. Rev. A Vol. 53, pp. 2038-2045(1996); H. E. Brandt, “Probe optimization in four-state protocol ofquantum cryptography,” Phys. Rev. A Vol. 66, 032303-1-16 (2002); H. E.Brandt, “Secrecy capacity in the four-state protocol of quantum keydistribution,” J. Math. Phys. Vol. 43, pp. 4526-4530 (2002); H. E.Brandt, “Optimization problem in quantum cryptography,” J. Optics B Vol.5, S557-560 (2003); H. E. Brandt, “Optimum probe parameters forentangling probe in quantum key distribution,” Quantum InformationProcessing Vol. 2, pp. 37-79 (2003); and H. E. Brandt, “Optimizedunitary transformation for BB84 entangling probe,” SPIE Proc., Vol.5436, pp. 48-64 (2004), all of which are incorporated herein byreference in their entirety. It should be noted that in Equation (132)of Brandt (2004), sin μ and cos μ should be interchanged in thecoefficient of |w₃

only. Also, e_(2θ) should be e_(θ). In Equation (195), the overall signof the coefficient of |w₂

should be ∓, rather than ±.

A complete optimization was performed by Brandt (“Probe optimization infour-state protocol of quantum cryptography,” 2002, “Optimizationproblem in quantum cryptography,” 2003, “Optimum probe parameters forentangling probe in quantum key distribution,” 2003, and “Optimizedunitary transformation for BB84 entangling probe,” 2004 above), in whichthree previously unknown sets of optimum probe parameters were obtained,all yielding the identical maximum information gain by the probe. Theprobe optimizations were based on maximizing the Rényi information gainby the probe on corrected data for a given error rate induced by theprobe in the legitimate receiver. A minimum overlap of the probe stateswhich are correlated with the signal states (because of theentanglement) determines the maximum Rényi information gain by theprobe. This is related to the idea that the more nearly orthogonal thecorrelated states are, the easier they are to distinguish. The upperbound on Rényi information gain by the probe is needed to calculate thesecrecy capacity of the BB84 protocol and to determine the number ofbits which must be sacrificed during privacy amplification in order thatit be exponentially unlikely that more than token leakage of the finalkey be available to the eavesdropper following key distillation (seeBrandt, “Secrecy capacity in the four-state protocol of quantum keydistribution,” 2002).

Using the simplest optimal set of probe parameters, it was shown byBrandt (2004) that the above unitary transformation representing theprobe produces the following entanglements for initial probe state |w

and incoming BB84 signal states |u

, |ū

, |v

, or | v

, respectively:

$\begin{matrix}{{{\left. u \right\rangle \otimes \left. w \right\rangle}->{\frac{1}{4}\left( {{\left. \alpha_{+} \right\rangle \otimes \left. u \right\rangle} + {\left. \alpha \right\rangle \otimes \left. \overset{\_}{u} \right\rangle}} \right)}},} & (1) \\{{{\left. \overset{\_}{u} \right\rangle \otimes \left. w \right\rangle}->{\frac{1}{4}\left( {{\left. \alpha \right\rangle \otimes \left. u \right\rangle} + {\left. \alpha_{-} \right\rangle \otimes \left. \overset{\_}{u} \right\rangle}} \right)}},} & (2) \\{{{\left. v \right\rangle \otimes \left. w \right\rangle}->{\frac{1}{4}\left( {{\left. \alpha_{-} \right\rangle \otimes \left. v \right\rangle} - {\left. \alpha \right\rangle \otimes \left. \overset{\_}{v} \right\rangle}} \right)}},} & (3) \\{{{\left. \overset{\_}{v} \right\rangle \otimes \left. w \right\rangle}->{\frac{1}{4}\left( {{{- \left. \alpha \right\rangle} \otimes \left. v \right\rangle} + {\left. \alpha_{+} \right\rangle \otimes \left. \overset{\_}{v} \right\rangle}} \right)}},} & (4)\end{matrix}$in which the probe states |α₊

, |α⁻

, and |α

are given by

$\begin{matrix}{{\left. \alpha_{+} \right\rangle = {{\begin{bmatrix}{{\left( {2^{1/2} + 1} \right)\left( {1 \pm \eta} \right)^{1/2}} +} \\{\left( {2^{1/2} - 1} \right)\left( {1 \mp \eta} \right)^{1/2}}\end{bmatrix}\left. w_{0} \right\rangle} + {\begin{bmatrix}{{\left( {2^{1/2} + 1} \right)\left( {1 \mp \eta} \right)^{1/2}} +} \\{\left( {2^{1/2} - 1} \right)\left( {1 \pm \eta} \right)^{1/2}}\end{bmatrix}\left. w_{3} \right\rangle}}},} & (5) \\{{\left. \alpha_{-} \right\rangle = {{\begin{bmatrix}{{\left( {2^{1/2} - 1} \right)\left( {1 \pm \eta} \right)^{1/2}} +} \\{\left( {2^{1/2} + 1} \right)\left( {1 \mp \eta} \right)^{1/2}}\end{bmatrix}\left. w_{0} \right\rangle} + {\begin{bmatrix}{{\left( {2^{1/2} - 1} \right)\left( {1 \mp \eta} \right)^{1/2}} +} \\{\left( {2^{1/2} + 1} \right)\left( {1 \pm \eta} \right)^{1/2}}\end{bmatrix}\left. w_{3} \right\rangle}}},} & (6) \\{{\left. \alpha \right\rangle = {{\left\lbrack {{- \left( {1 \pm \eta} \right)^{1/2}} + \left( {1 \mp \eta} \right)^{1/2}} \right\rbrack\left. w_{0} \right\rangle} + {\left\lbrack {{- \left( {1 \mp \eta} \right)^{1/2}} + \left( {1 \pm \eta} \right)^{1/2}} \right\rbrack\left. w_{3} \right\rangle}}},\mspace{20mu}{where}} & (7) \\{\mspace{20mu}{{\eta \equiv \left\lbrack {8{E\left( {1 - {2E}} \right)}} \right\rbrack^{1/2}},}} & (8)\end{matrix}$expressed in terms of the probe basis states |w₀

and |w₃

, and also the set error rate E induced by the probe. Note that theHilbert space of the probe is two-dimensional, depending on the twoprobe basis vectors, |w₀

and |w₃

. Also, it is assumed that E≦¼ here and throughout. Larger induced errorrates are presumably impractical. It is important to note here that inEquations (29) and (32) of Brandt (2004), the overall sign must bepositive in order to yield Equation (19) of Brandt (2004). Also inEquations (5) through (7) above, the sign choices correspond to theimplementation chosen here.

It is to be noted in Equation (1) that the projected probe state |ψ_(uu)

correlated with the correct received signal state, see Slutsky (1998)and Brandt (“Optimum probe parameters for entangling probe in quantumkey distribution,” 2003), in which the state |u

is sent by the transmitter, and is also received by the legitimatereceiver, is |α₊

. Analogously, using Equation (2), it follows that the correlated probestate |ψ_(ūū)

is |α⁻

. The two states |α₊

and |α⁻

are to be distinguished by the measurement of the probe. Also, accordingto Equations (3) and (4), the same two probe states |α₊

and |α⁻

are the appropriate correlated states |ψ v v

and |ψ_(vv)

, respectively. This is consistent with the assumption in Section II ofSlutsky (1998) that only two probe states must be distinguished by theprobe.

As a basis for the present disclosure, this two-dimensional optimizedunitary transformation, Equations (1) through (4), is used to show thata simple quantum circuit representing the optimal entangling probeconsists of a single CNOT gate, see M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information, Cambridge University Press(2000), which is incorporated herein by reference in its entirety, inwhich the control qubit consists of two polarization basis states of thesignal, the target qubit consists of two probe basis states, and theinitial state of the probe is set in a specific way by the error rate. Amethod is determined below for measuring the appropriate correlatedstates of the probe, and a design for the entangling probe is described.

The present disclosure is a design implementation of an entangling probewhich optimally entangles itself with the signal so as to obtain themaximum information on the pre-privacy amplified key in the BB84protocol of quantum key distribution.

The quantum circuit model of quantum computation is exploited todetermine the quantum circuit corresponding to the optimum unitarytransformation, Equations (1) through (4). It was shown in Brandt (2004)that the tensor products of the initial state |w

of the probe with the orthonormal basis states |e₀

and |e₁

of the signal transform as follows (See Equation (1) and Equations (35)through (40) of Brandt (2004)), again with the positive over-all signchoice in Equation (26) of Brandt (2004):|e ₀{circle around (×)}w

→|e ₀

{circle around (×)}|A ₁

  (9)and|e ₁{circle around (×)}w

→|e ₁

{circle around (×)}|A ₂

,  (10)expressed in terms of probe states |A₁

and |A₂

, where|A ₁

=α₁ |w ₀

+α₂ |w ₃

,  (11)|A ₂

=α₂ |w ₀

+α₁ |w ₃

,  (12)in whichα₁=2^(−1/2)(1±η)^(1/2),  (13)α₂=2^(−1/2)(1∓η)^(1/2),  (14)and η is given by Equation (8).

In the two-dimensional Hilbert space of the signal, the two orthogonalbasis states |e₀

and |e₁

are oriented symmetrically about the signal states |u

and |v

, and make angles of π/8 with the signal states |u

and |v

, respectively, e.g., Slutsky (1998). Next, consider a quantumcontrolled-not gate (CNOT gate), in which the control qubit consists ofthe two signal basis states {|e₀

, |e₁

}, and the target qubit consists of the probe basis states {|w₀

, |w₃

}, and such that when |e₀

enters the control port then {|w₀

, |w₃

} becomes {|w₃

, |w₀

} at the target output port, or when |e₁

enters the control port then {|w₀

, |w₃

} remains unchanged. It then follows that a quantum circuit affectingthe transformations (9) and (10), and thereby faithfully representingthe entangling probe, consists of this CNOT gate with the state |A₂

always entering the target port, and {|e₀

, |e₁

} entering the control port. When |e₀

enters the control port, then |A₂

becomes |A₁

, or when |e₁

enters the control port then |A₂

remains unchanged, in agreement with Equations (9) and (10) with |w

=|A₂

. According to the quantum circuit model of quantum computation, it isknown that three CNOT gates are in general necessary and sufficient inorder to implement an arbitrary number of unitary transformations of twoqubits, G. Vidal and C. M. Dawson, “Universal quantum circuit fortwo-qubit transformations with three controlled-NOT gates,” Phys. Rev. AVol. 69, 010301-1-4 (2004), which is incorporated herein by reference inits entirety. In the present case, a single CNOT gate suffices tofaithfully represent the optimized unitary transformation.

Next expanding the signal state |u

in terms of the signal basis states, using Equation (1) of Slutsky(1998), one has

$\begin{matrix}{\left. u \right\rangle = {{\cos\frac{\pi}{8}\left. e_{0} \right\rangle} + {\sin\frac{\pi}{8}\left. e_{1} \right\rangle}}} & (15) \\{{\left. \overset{\_}{u} \right\rangle \equiv {{{- \sin}\frac{\pi}{8}\left. e_{0} \right\rangle} + {\cos\frac{\pi}{8}\left. e_{1} \right\rangle}}},} & (16) \\{{\left. v \right\rangle \equiv {{\sin\frac{\pi}{8}\left. e_{0} \right\rangle} + {\cos\frac{\pi}{8}\left. e_{1} \right\rangle}}},} & (17) \\{\left. \overset{\_}{v} \right\rangle \equiv {{\cos\frac{\pi}{8}\left. e_{0} \right\rangle} - {\sin\frac{\pi}{8}{\left. e_{1} \right\rangle.}}}} & (18)\end{matrix}$It then follows from Equations (9), (10), and (15) that the CNOT gateaffects the following transformation when the signal state |u

enters the control port:

$\begin{matrix}{{{\left. u \right\rangle \otimes \left. A_{2} \right\rangle}->{{\cos\frac{\pi}{8}{\left. e_{0} \right\rangle \otimes \left. A_{1} \right\rangle}} + {\sin\frac{\pi}{8}{\left. e_{1} \right\rangle \otimes \left. A_{2} \right\rangle}}}},} & (19)\end{matrix}$Using Equations (15) through (18), one also has

$\begin{matrix}{{\left. e_{0} \right\rangle = {{\cos\frac{\pi}{8}\left. u \right\rangle} - {\sin\frac{\pi}{8}\left. \overset{\_}{u} \right\rangle}}},} & (20) \\{{\left. e_{1} \right\rangle = {{\cos\frac{\pi}{8}\left. v \right\rangle} - {\sin\frac{\pi}{8}\left. \overset{\_}{v} \right\rangle}}},} & (21)\end{matrix}$Next substituting Equations (20) and (21) in Equation (19), one has

$\begin{matrix}{{\left. u \right\rangle \otimes \left. A_{2} \right\rangle}->{\cos\frac{\pi}{8}\begin{matrix}{{\left( {{\cos\frac{\pi}{8}\left. u \right\rangle} - {\sin\frac{\pi}{8}\left. \overset{\_}{u} \right\rangle}} \right) \otimes \left. A_{1} \right\rangle} +} \\{{\sin\frac{\pi}{8}{\left( {{\cos\frac{\pi}{8}\left. v \right\rangle} - {\sin\frac{\pi}{8}\left. \overset{\_}{v} \right\rangle}} \right) \otimes \left. A_{2} \right\rangle}},}\end{matrix}}} & (22) \\{{and}\mspace{14mu}{using}} & \; \\{{{\sin\frac{\pi}{8}} = {\frac{1}{2}\left( {2 - 2^{1/2}} \right)^{1/2}}},} & (23) \\{{{\cos\frac{\pi}{8}} = {\frac{1}{2}\left( {2 + 2^{1/2}} \right)^{1/2}}},} & (24)\end{matrix}$then Equation (22) becomes

$\begin{matrix}{{{\left. u \right\rangle \otimes \left. A_{2} \right\rangle}->{\frac{1}{4}\begin{bmatrix}{{\left( {2 + 2^{1/2}} \right){\left. A_{1} \right\rangle \otimes \left. u \right\rangle}} - {2^{1/2}{\left. A_{1} \right\rangle \otimes \left. \overset{\_}{u} \right\rangle}} +} \\{{2^{1/2}{\left. A_{2} \right\rangle \otimes \left. v \right\rangle}} - {\left( {2 - 2^{1/2}} \right){\left. A_{2} \right\rangle \otimes \left. \overset{\_}{v} \right\rangle}}}\end{bmatrix}}},} & (25)\end{matrix}$From Equations (15) through (18) and Equations (20), (21), (23), and(24), it follows that|v

=2^(−1/2) |u

+2^(−1/2) |ū

,  (26)| v

=2^(−1/2) |u

−2^(−1/2) |ū

,  (27)|u

=2^(−1/2) |v

+2^(−1/2) | v

,  (28)|ū

=2^(−1/2) |v

−2^(−1/2) | v

.  (29)Then substituting Equations (26) and (27) in Equation (25), and usingEquations (5), and (6), one obtains Equation (1). Analogously, one alsoobtains Equations (2) through (4).

One concludes that the quantum circuit consisting of the CNOT gate doesin fact faithfully represent the action of the optimum unitarytransformation in entangling the signal states |u

, |ū

, |v

, and | v

with the probe states |α₊

, |α⁻

, and |α

. It is to be emphasized that the initial state of the probe must be |A₂

, given by Equation (12). (A sign choice in Equations (13) and (14) ismade below, consistent with the measurement procedure defined there.)

According to Equations (1) through (4), and the above analysis, theprobe produces the following entanglements for initial probe state |w

=|A₂

and incoming signal states |u

, |ū

, |v

, or | v

, respectively:

$\begin{matrix}{{{\left. u \right\rangle \otimes \left. A_{2} \right\rangle}->{\frac{1}{4}\left( {{\left. \alpha_{+} \right\rangle \otimes \left. u \right\rangle} + {\left. \alpha \right\rangle \otimes \left. \overset{\_}{u} \right\rangle}} \right)}},} & (30) \\{{{\left. \overset{\_}{u} \right\rangle \otimes \left. A_{2} \right\rangle}->{\frac{1}{4}\left( {{\left. \alpha \right\rangle \otimes \left. u \right\rangle} + {\left. \alpha_{-} \right\rangle \otimes \left. \overset{\_}{u} \right\rangle}} \right)}},} & (31) \\{{{\left. v \right\rangle \otimes \left. A_{2} \right\rangle}->{\frac{1}{4}\left( {{\left. \alpha_{-} \right\rangle \otimes \left. v \right\rangle} - {\left. \alpha \right\rangle \otimes \left. \overset{\_}{v} \right\rangle}} \right)}},} & (32) \\{{\left. \overset{\_}{v} \right\rangle \otimes \left. A_{2} \right\rangle}->{\frac{1}{4}{\left( {{{- \left. \alpha \right\rangle} \otimes \left. v \right\rangle} + {\left. \alpha_{+} \right\rangle \otimes \left. \overset{\_}{v} \right\rangle}} \right).}}} & (33)\end{matrix}$Then, according to Equations (30) and (31), if, following the publicreconciliation phase of the BB84 protocol, the signal basis mutuallyselected by the legitimate transmitter and receiver is publicly revealedto be {|u

, |ū

}, then the probe measurement must distinguish the projected probe state|α₊

, when the signal state |u

is both sent and received, from the projected probe state |α⁻

, when the signal state |ū

is both sent and received. In this case one has the correlations:|u

|α₊

,  (34)|ū

|α⁻

.  (35)The same two states |α₊

and |α⁻

must be distinguished, no matter which basis is chosen duringreconciliation. Thus, according to Equations (32) and (33), if,following the public reconciliation phase of the BB84 protocol, thesignal basis mutually selected by the legitimate transmitter andreceiver is publicly revealed to be {|v

, | v

}, then the probe measurement must distinguish the projected probe state|α⁻

, when the signal state |v

is both sent and received, from the projected probe state |α₊

, when the signal state | v

is both sent and received. In this case one has the correlations:|v

|α⁻

,  (36)| v

|α₊

.  (37)

Next, one notes that the correlations of the projected probe states |α₊

and |α⁻

with the probe's two orthogonal basis states |w₀

and |w₃

are indicated, according to Equations (5) and (6), by the followingprobabilities:

$\begin{matrix}\begin{matrix}{\frac{{\left\langle {w_{0}❘\alpha_{+}} \right\rangle }^{2}}{{\alpha_{+}}^{2}} = \frac{{\left\langle {w_{3}❘\alpha_{-}} \right\rangle }^{2}}{{\alpha_{-}}^{2}}} \\{{= {\frac{1}{2} \pm \frac{\left\lbrack {E\left( {1 - {2E}} \right)} \right\rbrack^{1/2}}{\left( {1 - E} \right)}}},}\end{matrix} & (38) \\\begin{matrix}{\frac{{\left\langle {w_{0}❘\alpha_{-}} \right\rangle }^{2}}{{\alpha_{-}}^{2}} = \frac{{\left\langle {w_{3}❘\alpha_{+}} \right\rangle }^{2}}{{\alpha_{+}}^{2}}} \\{= {\frac{1}{2} \mp {\frac{\left\lbrack {E\left( {1 - {2E}} \right)} \right\rbrack^{1/2}}{\left( {1 - E} \right)}.}}}\end{matrix} & (39)\end{matrix}$At this point, the positive sign is chosen in Equation (38), andcorrespondingly the negative sign in Equation (39). This choice servesto define the Hilbert-space orientation of the probe basis states, inorder that the probe basis state |w₀

be dominantly correlated with the signal states |u

and | v

, and that the probe basis state |w₃

be dominantly correlated with the signal states |ū

and |v

. With this sign choice, then Equations (38) and (39) become

$\begin{matrix}\begin{matrix}{\frac{{\left\langle {w_{0}❘\alpha_{+}} \right\rangle }^{2}}{{\alpha_{+}}^{2}} = \frac{{\left\langle {w_{3}❘\alpha_{-}} \right\rangle }^{2}}{{\alpha_{-}}^{2}}} \\{{= {\frac{1}{2} + \frac{\left\lbrack {E\left( {1 - {2E}} \right)} \right\rbrack^{1/2}}{\left( {1 - E} \right)}}},}\end{matrix} & (40) \\\begin{matrix}{\frac{{\left\langle {w_{0}❘\alpha_{-}} \right\rangle }^{2}}{{\alpha_{-}}^{2}} = \frac{{\left\langle {w_{3}❘\alpha_{+}} \right\rangle }^{2}}{{\alpha_{+}}^{2}}} \\{{= {\frac{1}{2} - \frac{\left\lbrack {E\left( {1 - {2E}} \right)} \right\rbrack^{1/2}}{\left( {1 - E} \right)}}},}\end{matrix} & (41)\end{matrix}$and one then has the following state correlations.

$\begin{matrix}{\left. \left. \alpha_{+} \right\rangle\Leftrightarrow\left. w_{0} \right\rangle \right.,} & (42) \\\left. \left. \alpha_{-} \right\rangle\Leftrightarrow{\left. w_{3} \right\rangle.} \right. & (43)\end{matrix}$Next combining the correlations (36), (37), (42), and (43), one thenestablishes the following correlations:

$\begin{matrix}{\left. \left\{ {\left. u \right\rangle,\left. \overset{\_}{v} \right\rangle} \right\}\Leftrightarrow\left. \alpha_{+} \right\rangle\Leftrightarrow\left. w_{0} \right\rangle \right.,} & (44) \\{\left. \left\{ {\left. \overset{\_}{u} \right\rangle,\left. v \right\rangle} \right\}\Leftrightarrow\left. \alpha_{-} \right\rangle\Leftrightarrow\left. w_{3} \right\rangle \right.,} & (45)\end{matrix}$to be implemented by the probe measurement method. This can be simplyaccomplished by a von Neumann-type projective measurement of theorthogonal probe basis states |w₀

and |w₃

, implementing the probe projective measurement operators {|w₀><w₀|,|w₃><w₃|}. The chosen geometry in the two-dimensional Hilbert space ofthe probe is such that the orthogonal basis states |w₀

and |w₃

make equal angles with the states |α₊

and |α⁻

, respectively, and the sign choice is enforced in Equations (5) and(6), namely,

$\begin{matrix}{{\left. \alpha_{+} \right\rangle = {{\begin{bmatrix}{{\left( {2^{1/2} + 1} \right)\left( {1 + \eta} \right)^{1/2}} +} \\{\left( {2^{1/2} - 1} \right)\left( {1 - \eta} \right)^{1/2}}\end{bmatrix}\left. w_{0} \right\rangle} + {\begin{bmatrix}{{\left( {2^{1/2} + 1} \right)\left( {1 - \eta} \right)^{1/2}} +} \\{\left( {2^{1/2} - 1} \right)\left( {1 + \eta} \right)^{1/2}}\end{bmatrix}\left. w_{3} \right\rangle}}},} & (46) \\{{\left. \alpha_{-} \right\rangle = {{\begin{bmatrix}{{\left( {2^{1/2} + 1} \right)\left( {1 - \eta} \right)^{1/2}} +} \\{\left( {2^{1/2} - 1} \right)\left( {1 + \eta} \right)^{1/2}}\end{bmatrix}\left. w_{0} \right\rangle} + {\begin{bmatrix}{{\left( {2^{1/2} + 1} \right)\left( {1 + \eta} \right)^{1/2}} +} \\{\left( {2^{1/2} - 1} \right)\left( {1 - \eta} \right)^{1/2}}\end{bmatrix}\left. w_{3} \right\rangle}}},} & (47) \\\text{where} & \; \\{{\eta = \left\lbrack {8{E\left( {1 - {2E}} \right)}} \right\rbrack^{1/2}},} & (48)\end{matrix}$as in Equation (8). This geometry is consistent with the symmetric vonNeumann test, which is an important part of the optimization.

An object of the present disclosure is to provide an eavesdropping probefor obtaining maximum information on the pre-privacy-amplified key inthe BB84 protocol of quantum key distribution. In a preferredembodiment, the device is a probe that entangles itself separately witheach signal photon on its way between the legitimate sender andreceiver, in such a way as to obtain maximum information on thepre-privacy-amplified key.

An incident photon coming from the legitimate transmitter is received bythe probe in one of the four signal-photon linear-polarization states |u

, |ū

, |v

, or | v

in the BB84 protocol. The signal photon enters the control port of theCNOT gate. The initial state of the probe is a photon inlinear-polarization state |A₂

and entering the target port of the CNOT gate. The probe photon isproduced by a single-photon source and is appropriately timed withreception of the signal photon by first sampling a few successive signalpulses to determine the repetition rate of the transmitter. Thelinear-polarization state |A₂

, according to Equations (12) through (14) and Equation (8), with thesign choice made above, is given by

$\begin{matrix}{\left. A_{2} \right\rangle = \begin{matrix}{{\left\lbrack {\frac{1}{2}\left\{ {1 - \left\lbrack {8{E\left( {1 - {2E}} \right)}} \right\rbrack^{1/2}} \right\}} \right\rbrack^{1/2}\left. w_{0} \right\rangle} +} \\{{\left\lbrack {\frac{1}{2}\left\{ {1 + \left\lbrack {8{E\left( {1 - {2E}} \right)}} \right\rbrack^{1/2}} \right\}} \right\rbrack^{1/2}\left. w_{3} \right\rangle},}\end{matrix}} & (49)\end{matrix}$and can be simply set for an error rate E by means of a polarizer. Inthis way the device can be tuned to the chosen error rate (E≦¼) inducedby the probe. The outgoing gated signal photon is relayed on to thelegitimate receiver, and the gated probe photon enters a Wollastonprism, oriented to separate orthogonal photon linear-polarization states|w₀

and |w₃

, and the photon is then detected by one of two photodetectors. If thebasis, revealed during the public basis-reconciliation phase of the BB84protocol, is {|u

, |ū

}, then the photodetector located to receive the polarization state |w₀

or |w₃

, respectively, will indicate, in accord with the correlations (44) and(45), that a state |u

or |ū

, respectively, was most likely measured by the legitimate receiver.Alternatively, if the announced basis is {|v

, | v

}, then the photodetector located to receive the polarization state |w₃

or |w₀

, respectively, will indicate, in accord with the correlations (44) and(45), that a state |v

or | v

, respectively, was most likely measured by the legitimate receiver. Bycomparing the record of probe photodetector triggering with the sequenceof bases revealed during reconciliation, then the likely sequence ofones and zeroes constituting the key, prior to privacy amplification,can be assigned. In any case the net effect is to yield, for a set errorrate E, the maximum information gain to the probe, which is given byEquation (19) of Brandt (2004), namely,

$\begin{matrix}{I_{opi}^{R} = {{\log_{2}\left\lbrack {2 - \left( \frac{1 - {3E}}{1 - E} \right)^{2}} \right\rbrack}.}} & (50)\end{matrix}$

The geometry of the initial and shifted probe polarization states |A₂

and |A₁

, respectively, and probe basis states, |w₀

and |w₃

, in the two-dimensional Hilbert space of the probe is such that theangle δ₀ between the probe state |A₁

and the probe basis state |w₀

is given by

$\begin{matrix}{{\delta_{0} = {\cos^{- 1}\left( \frac{\left\langle w_{0} \right.A_{1}}{A_{1}} \right)}},} & (51)\end{matrix}$or, substituting |A₁

, given by Equations (11), (13), and (14) with the sign choice made inEquation (49), namely,

$\begin{matrix}{\left. A_{1} \right\rangle = \begin{matrix}{{\left\lbrack {\frac{1}{2}\left\{ {1 + \left\lbrack {8{E\left( {1 - {2E}} \right)}} \right\rbrack^{1/2}} \right\}} \right\rbrack^{1/2}\left. w_{0} \right\rangle} +} \\{{\left\lbrack {\frac{1}{2}\left\{ {1 - \left\lbrack {8{E\left( {1 - {2E}} \right)}} \right\rbrack^{1/2}} \right\}} \right\rbrack^{1/2}\left. w_{3} \right\rangle},}\end{matrix}} & (52)\end{matrix}$in Equation (51), one obtains

$\begin{matrix}{\delta_{0} = {{\cos^{- 1}\left( {\frac{1}{2}\left\{ {1 + \left\lbrack {8{E\left( {1 - {2E}} \right)}} \right\rbrack^{1/2}} \right\}} \right)}^{1/2}.}} & (53)\end{matrix}$This is also the angle between the initial linear-polarization state |A₂

of the probe and the probe basis state |w₃

. Also, the shift δ in polarization between the initial probe states |A₂

and the state |A₁

(the angle between |A₁

and |A₂

) is given by

$\begin{matrix}{{\delta = {\cos^{- 1}\left( \frac{\left\langle {A_{1}❘A_{2}} \right\rangle}{{A_{1}}{A_{2}}} \right)}},} & (54)\end{matrix}$or, substituting Equations (49) and (52), one obtainsδ=cos⁻¹(1−4E).  (55)Possible implementations of the CNOT gate may include ones based oncavity-QED, solid state, and/or linear optics.

FIG. 3 shows a system for obtaining information on a key in the BB84protocol of quantum key distribution. The embodiment in FIG. 3 is acircuit design for a quantum cryptographic entangling probe 300. Thestraight lines with arrows represent possible optical pathways (inoptical fiber or in free space) of the signal photon 140 and the probephoton to move through the quantum cryptographic entangling probe 300.The path labeled S1 is the incoming path for a signal photon 140 fromthe transmitter 110 in one of four possible BB84 signal states {|u

, |ū

, |v

, | v

}. The path labeled S2 is the path of the gated signal photon 160 on itsway to the legitimate receiver 120. The path labeled P1 is the path ofthe probe photon produced by a single photon source 320 and passingthrough a set polarization filter (linear polarizer) 330 prior toentering the target port of the CNOT gate 310. The path P2 is that ofthe gated probe photon 150 on its way to the Wollaston prism 340, andthe paths P3 and P4 are possible paths of the gated probe photon 150from the Wollaston prism 340 to the photodetectors 350 and 360.

The quantum circuit, faithfully representing the optimum entanglingprobe, consists of a single quantum-controlled not gate (CNOT gate) 310in which the control qubit consists of two photon-polarization basisstates of the signal, the target qubit consists of the two probe-photonpolarization basis states, and the probe photon is prepared in theinitial linear-polarization state, Equation (49), set by the inducederror rate. The initial polarization state of the probe photon can beproduced by a single-photon source 320 together with a linear polarizer330. The probe photon can be appropriately timed with reception of asignal photon by first sampling a few successive signal pulses todetermine the repetition rate of the transmitter 110. The gated probephoton 150, optimally entangled with the signal, enters a Wollastonprism 340, which separates the appropriate correlated states of theprobe photon to trigger one or the other of two photodetectors 350 or360. Basis selection, revealed on the public channel during basisreconciliation in the BB84 protocol, is exploited to correlatephotodetector clicks with the signal transmitting the key, and to assignthe most likely binary numbers, 1 or 0, such that the information gainby the quantum cryptographic entangling probe 300 of the key, prior toprivacy amplification, is maximal. Explicit design parameters for theentangling probe are analytically specified, including: (1) the explicitinitial polarization state of the probe photon, Equation (49); (2) thetransition state of the probe photon, Equation (52); (3) theprobabilities that one or the other photodetector triggers correspondingto a 0 or 1 of the key, Equations (40) and (41); (4) the relative anglesbetween the various linear-polarization states in the Hilbert space ofthe probe, Equations (53) and (55); and (5) the information gain by theprobe, Equation (50).

The quantum cryptographic entangling probe 300 is a simplespecial-purpose quantum information processor that will improve the oddsfor an eavesdropper in gaining access to the pre-privacy-amplified key,as well as imposing a potentially severe sacrifice of key bits duringprivacy amplification (Brandt, “Secrecy capacity in the four-stateprotocol of quantum key distribution,” 2002). The quantum cryptographicentangling probe 300 measures the maximum information on the pre-privacyamplified key in the BB84 protocol of quantum key distribution.

It should be emphasized that the above-described embodiments of thepresent disclosure, are merely possible examples of implementations,merely set forth for a clear understanding of the principles of thedisclosure. Many variations and modifications may be made to theabove-described embodiment(s) of the disclosure without departingsubstantially from the spirit and principles of the disclosure. All suchmodifications and variations are intended to be included herein withinthe scope of this disclosure.

1. A system for eavesdropping on a four state quantum key distributionprotocol, comprising: providing a quantum cryptographic entangling probecomprising: a single-photon probe creator for the purpose ofeavesdropping on quantum key distribution; a photon polarization statefilter to tune the polarization state of the probe photon produced bysaid single photon probe creator to the photon polarization state for aset error rate induced by the quantum cryptographic entangling probe; aquantum controlled-NOT (CNOT) gate configured to provide entanglement ofa signal photon polarization state with the probe photon polarizationstate and produce a gated probe photon so as to obtain information on akey; a Wollaston prism in communication with said CNOT gate configuredto separate the gated probe photon with polarization correlated to asignal measured by a legitimate receiver; and two single-photon photodetectors for the purpose of eavesdropping on quantum key distributionin communication with said Wollaston prism configured to measure thepolarization state of the gated probe photon.
 2. The system of claim 1,wherein the quantum CNOT gate is further configured to provide optimumentanglement of the signal with the probe photon polarization state toproduce the gated probe photon so as to obtain maximum Renyi informationfrom the signal.
 3. The system of claim 1, wherein the signalpolarization state remains substantially unaffected.
 4. The system ofclaim 1, wherein the CNOT gate is one of a quantum dot implementation ora cavity QED implementation.
 5. The system of claim 1, wherein the CNOTgate is a solid state implementation.
 6. The system of claim 1, whereinthe CNOT gate is a linear optics implementation.
 7. A method forobtaining eavesdropping information on a four state quantum keydistribution protocol, said method comprising: providing a single-photonprobe producer; inducing a polarization state corresponding to a seterror rate; interacting a probe photon with a signal state and producinga gated probe photon with a polarization state so as to obtaininformation on a key; photodetecting the gated probe photon withpolarization correlated with a signal measured by a legitimate receiver;extracting information on polarization-basis selection available on apublic classical communication channel between the transmitter and thelegitimate receiver; and determining the polarization state measured bythe probe receiver.
 8. The method of claim 7, wherein the interactingcomprises entangling the signal with the probe photon polarization stateso as to obtain maximum Renyi information from the signal.
 9. The methodof claim 7, further comprising receiving a signal photon from atransmitter for the purpose of timing prior to the configuring of thesingle-photon source.
 10. The method of claim 7, further comprisingrelaying an outgoing gated signal photon to the receiver.
 11. The methodof claim 7, wherein the probe receiver utilizes a Wollaston prism. 12.The method of claim 7, wherein the probe receiver utilizes apolarization beam splitter.
 13. The method of claim 7, wherein thesingle-photon is timed by receiving a signal photon from a transmitterby first sampling successive signal pulses to determine a repetitionrate of the transmitter.
 14. A system for obtaining eavesdroppinginformation on a four state quantum key distribution protocol,comprising: means for producing a single probe photon; means fordetermining a polarization state corresponding to a set error rate;means for correlating the probe photon polarization state with at leastone photon from the quantum key; means for routing the probe photon;means for photo detecting the probe photon; and means for extractinginformation available on a public classical communication channelbetween the transmitter and the receiver.
 15. The system of claim 14,further comprising means for receiving a signal photon from atransmitter.
 16. The system of claim 14, further comprising means forrelaying an outgoing signal photon to a receiver.
 17. The system ofclaim 14, further comprising means for providing entanglement of thesignal with the probe photon polarization state so as to obtain maximumRenyi information from the signal.
 18. The system of claim 14, furthercomprising means for timing the probe single-photon by receiving asignal photon from a transmitter by first sampling successive signalpulses to determine a repetition rate of the transmitter.
 19. The systemof claim 14 wherein the means for correlating comprises a CNOT gate. 20.The system of claim 14 wherein the polarization states of the photonstransmitting the quantum key are detected by the means for correlatingwithout substantially effecting the photons in the quantum key such thatthe intended recipient of the quantum key is unaware of the detection.